Nonlinear dynamics and pattern bifurcations in a model for …jas/talks/beijing_jun09.pdf ·...

Ecological BackgroundThe Mathematical Model

Linear AnalysisTravelling Wave Equations

Pattern StabilityConclusions

Nonlinear dynamics and pattern bifurcationsin a model for vegetation stripes

in arid environments

Jonathan A. Sherratt

Department of MathematicsHeriot-Watt University

Beijing, June 2009

This talk can be downloaded from my web sitewww.ma.hw.ac.uk/∼jas

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Nonlinear Dynamics of Vegetation Stripes




In collaboration withGabriel Lord





Outline

1 Ecological Background

2 The Mathematical Model

3 Linear Analysis

4 Travelling Wave Equations

5 Pattern Stability

6 Conclusions





Vegetation Pattern FormationMosaic and Striped PatternsMechanisms for Vegetation Patterning

Outline



3 Linear Analysis


5 Pattern Stability

6 Conclusions






Vegetation Pattern Formation

Vegetation patterns are found in semi-arid areas of Africa,Australia and MexicoFirst identified by aerial photos in 1950sPlants vary from grasses to shrubs and trees






Mosaic and Striped Patterns

Bushy vegetation in Niger

Mitchell grass in Australia(Western New South Wales)






Mosaic and Striped Patterns

On flat ground, irregularmosaics of vegetationare typical

On slopes, the patternsare stripes, parallel tocontours (“Tiger bush”)






Mechanisms for Vegetation Patterning

Basic mechanism: competition for water








Possible detailed mechanism: water flow downhill causesstripes

FLOW

WATER

FLOW

WATER









































































































This mechanism suggests that the stripes move uphill





Mathematical Model of KlausmeierTypical Solution of the Model

Outline



3 Linear Analysis


5 Pattern Stability

6 Conclusions






Mathematical Model of Klausmeier

Rate of change = Rainfall – Evaporation – Uptake by + Flowof water plants downhill

Rate of change = Growth, proportional – Mortality +Randomplant biomass to water uptake dispersal

∂w/∂t = A − w − wu2 + ν∂w/∂x

∂u/∂t = wu2 − Bu + ∂2u/∂x2









∂w/∂t = A − w − wu2 + ν∂w/∂x

∂u/∂t = wu2 − Bu + ∂2u/∂x2

The nonlinearity in wu2 arises because the presence of roots in-creases water infiltration into the soil.









∂w/∂t = A − w − wu2 + ν∂w/∂x

∂u/∂t = wu2 − Bu + ∂2u/∂x2

Parameters: A: rainfall B: plant loss ν: slope






Typical Solution of the Model















































































































































































































































Homogeneous Steady StatesApproximate Conditions for PatterningAn Illustration of Conditions for PatterningPredicting Pattern WavelengthShortcomings of Linear Stability Analysis

Outline



3 Linear Analysis


5 Pattern Stability

6 Conclusions






Homogeneous Steady States

For all parameter values, there is a stable “desert” steadystate u = 0, w = A.








When A ≥ 2B, there are also two non-trivial steady states

uu =2B

A +√

A2 − 4B2wu =

A +√

A2 − 4B2

2

us =2B

A −√

A2 − 4B2ws =

A −√

A2 − 4B2

2








When A ≥ 2B, there are also two non-trivial steady states

uu =2B

A +√

A2 − 4B2wu =

A +√

A2 − 4B2

2unstable

us =2B

A −√

A2 − 4B2ws =

A −√

A2 − 4B2

2stable to homogpertns for B < 2

Patterns develop when (us, ws) is unstable toinhomogeneous perturbations






Approximate Conditions for Patterning

Look for solutions (u, w) = (us, ws) + (u0, w0) exp{ikx + λt}

The dispersion relation Re[λ(k)] is algebraically complicated









An approximate condition for pattern formation isA < ν1/2 B5/4/ 81/4









An approximate condition for pattern formation is2B < A < ν1/2 B5/4/ 81/4

One can niavely assume that existence of (us, ws) gives asecond condition






An Illustration of Conditions for Patterning

The dots show parameters forwhich there are growinglinear modes.







The upper red lineis A = ν1/2 B5/4/ 81/4.

The lower red lineis A = 2B.







Numerical simulations showpatterns in both the dottedand green regions ofparameter space.






Predicting Pattern Wavelength

Pattern wavelength is the most accessible property ofvegetation stripes in the field, via aerial photography.Wavelength can be predicted from the linear analysis







Pattern wavelength is the most accessible property ofvegetation stripes in the field, via aerial photography.Wavelength can be predicted from the linear analysis

However this predictiondoesn’t fit the patternsseen in numericalsimulations.






Shortcomings of Linear Stability Analysis

Linear stability analysis fails in two ways:

It significantly over-estimates the minimum rainfall level forpatterns.

Close to the maximum rainfall level for patterns, itincorrectly predicts a variation in pattern wavelength withrainfall.





Travelling Wave EquationsBifurcation Diagram for Travelling Wave ODEsWhen do Patterns Form?Pattern Formation for Low Rainfall

Outline



3 Linear Analysis


5 Pattern Stability

6 Conclusions






Travelling Wave Equations

The patterns move at constant shape and speed⇒ u(x , t) = U(z), w(x , t) = W (z), z = x − ct

d2U/dz2 + c dU/dz + WU2 − BU = 0

(ν + c)dW/dz + A − W − WU2 = 0

The patterns are periodic (limit cycle) solutions of these ODEs






Bifurcation Diagram for Travelling Wave ODEs












When do Patterns Form?


















Pattern Formation for Low Rainfall

Patterns are also seen forparameters in the greenregion.

















Discretizing the PDEsBifurcation Diagram for Discretized PDEsSpeed vs Rainfall for Discretized PDEsKey ResultsHysteresis

Outline



3 Linear Analysis


5 Pattern Stability

6 Conclusions






Discretizing the PDEs

To investigate pattern stability, we must work with the modelPDEs. We discretize these in space and then use AUTO tostudy the resulting ODE system:

∂ui/∂t = wiu2i − Bui + (ui+1 − 2ui + 2ui-1)/∆x2

∂wi/∂t = A − wi − wiu2i + ν(wi+1 − wi)/∆x

(i = 1, . . . , N).










(i = 1, . . . , N).We use upwinding for the convective term.










(i = 1, . . . , N).We use upwinding for the convective term.Most of our work has used N = 40 and ∆x = 2.We assume periodic boundary conditions.






Bifurcation Diagram for Discretized PDEs






























Speed vs Rainfall for Discretized PDEs

c vs A for PDEs






Speed vs Rainfall for Discretized PDEs

c vs A for PDEs c vs A for travelling wave PDEs






Key Result

Many of the possible patterns areunstable and thus will never be seen.

However, for a wide range of rainfalllevels, there are multiple stablepatterns.






Hysteresis

Rai

nfal

l

Space

The existence of multiple stablepatterns raises the possibility ofhysteresis

We consider slow variations in therainfall parameter A

Parameters correspond to grass,and the rainfall range corresponds to130–930 mm/year






Hysteresis

Tim

eR

ainf

all

Space

<< Mode 5 >> <<<<< Mode 1 >>>>> < Mode 3 >






Hysteresis

Tim

eR

ainf

all

Space

<< Mode 5 >> <<<<< Mode 1 >>>>> < Mode 3 >





Predictions of Pattern WavelengthOther Potential Mechanisms for Vegetation PatternsReferences

Outline



3 Linear Analysis


5 Pattern Stability

6 Conclusions






Predictions of Pattern Wavelength

In general, pattern wavelength depends on initialconditions

When vegetation stripes arise from homogeneousvegetation via a decrease in rainfall, pattern wavelengthwill remain at its bifurcating value.






Predictions of Pattern Wavelength

In general, pattern wavelength depends on initialconditions

When vegetation stripes arise from homogeneousvegetation via a decrease in rainfall, pattern wavelengthwill remain at its bifurcating value.

Wavelength =

√

8π2

Bν






Other Potential Mechanisms for Vegetation Patterns

Rietkirk Klausmeier model with diffusion of water in the soil

van de Koppel Klausmeier model with grazing

Maron two variable model (plant density and water in thesoil) with water transport based on porous mediatheory

Lejeune short range activation (shading) and long rangeinhibition (competition for water)

All of these models predict patterns. To discriminate betweenthem requires a detailed understanding of each model.






References

J.A. Sherratt: An analysis of vegetation stripe formation insemi-arid landscapes. J. Math. Biol. 51, 183-197 (2005).

J.A. Sherratt, G.J. Lord: Nonlinear dynamics and patternbifurcations in a model for vegetation stripes in semi-aridenvironments. Theor. Pop. Biol. 71, 1-11 (2007).

These papers can be downloaded from my web sitewww.ma.hw.ac.uk/∼jas






List of Frames


Vegetation Pattern Formation

Mosaic and Striped PatternsMechanisms for Vegetation Patterning

2 The Mathematical ModelMathematical Model of KlausmeierTypical Solution of the Model

3 Linear AnalysisHomogeneous Steady States




Shortcomings of Linear Stability Analysis


Travelling Wave Equations


When do Patterns Form?Pattern Formation for Low Rainfall

5 Pattern StabilityDiscretizing the PDEs


Speed vs Rainfall for Discretized PDEsKey ResultsHysteresis

6 ConclusionsPredictions of Pattern Wavelength

Other Potential Mechanisms for Vegetation Patterns

References






Pattern Selection

For a range of rainfall levels, there is more than one stablepattern. Which will be selected?We consider initial conditions that are small perturbationsof the coexistence steady state (us, vs).All such initial conditions give a pattern, but the wavelengthdepends on the initial perturbation






Pattern Selection







Pattern Selection







Pattern Selection


The wavelengthis close to thatpredicted bylinear stabilityanalysis






Pattern Selection on Larger Domains

The proximity of thewavelength to the mostlinearly unstable modecontinues as thedomain is enlarged

10 11 12 13 140

50

100

150

200

250

300

350

400

450

Mode

Fre

quen

cy

A=1.8, L=320, dx=2






Pattern Selection on Larger Domains

The proximity of thewavelength to the mostlinearly unstable modecontinues as thedomain is enlarged

10 11 12 13 140

50

100

150

200

250

300

350

400

450

Mode

Fre

quen

cy

A=1.8, L=320, dx=2

But it does not apply for other initialconditions, such as perturbationsabout (uu, wu)


Date post:	15-Apr-2020
Category:	Documents
Upload:	others
View:	2 times
Download:	0 times

Nonlinear dynamics and pattern bifurcations in a model for …jas/talks/beijing_jun09.pdf ·...

Documents